MoL-2012-10:
Henk, Paula
(2012)
*Supremum in the Lattice of Interpretability.*
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## Abstract

This thesis is located in the field of provability and

interpretability logic, where modal logic is used in the study of

formal systems of arithmetic. The central notion of this thesis is

that of interpretability. The notion of interpretability can be seen

as a tool for comparing axiomatic theories. Intuitively, if a theory T

interprets a theory S , T is at least as strong as S. The modal logic

ILM captures exactly what Peano Arithemtic (PA) can prove about

interpretability between nite extensions of itself. As it turns out,

nite extensions of PA form a lattice under the relation of

interpretability, i.e. any two theories have an inmum and a supremum

in the interpretability ordering. The supremum in this lattice is the

main subject of study in this thesis.

We will extend the logic ILM with a binary operator for the supremum,

and explore the possibilities of having a modal semantics for the

resulting system ILMS. For that purpose, the supremum will be studied

both from the arithmetical as well as from the modal

perspective. First, we will see that the exact content of the logic

ILMS depends on the formula that is chosen as the arithmetical

representative of the supremum. This is dierent from ILM, where the

meaning of the modal symbols is xed from the outset. Proceeding to the

modal side, we establish an important negative result: there can be no

structural characterization of ILM{models that validate the dening

axiom for the supremum. This precludes the possibility of having a

relational semantics for the system ILMS | at least one that would

extend the usual semantics for ILM. Finally, we examine an elegant but

unfortunately failed attempt to nd a relational semantics for a

particular representative of the supremum.

Item Type: | Report |
---|---|

Report Nr: | MoL-2012-10 |

Series Name: | Master of Logic Thesis (MoL) Series |

Year: | 2012 |

Uncontrolled Keywords: | Logic, Mathematics |

Depositing User: | Tanja Kassenaar |

Date Deposited: | 12 Oct 2016 14:38 |

Last Modified: | 12 Oct 2016 14:38 |

URI: | https://eprints.illc.uva.nl/id/eprint/876 |

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